Chapter 10        The polar performance diagram explained using the swing rig version of the Bermuda rig


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   Diagram 10-1. 
The polar performance diagram

It seems to me that the key to an understanding of the sailing of racing yachts is the polar performance diagram. This diagram is just a set of arrows all starting from one point like the radii of a circle. They are often set at 10° intervals. The direction of each arrow represents a course sailed by the yacht in a true wind of constant speed and the length represents the measured speed of the yacht. In effect the diagram shows what speed the boat has achieved at any given point of sailing. To this limited extent it is a performance graph. I know of no way to make the measurements needed to draw such a diagram for a model yacht although it seems to be done regularly for full sized yachts. I need such a diagram at this point in the text and I have drawn it using the outcome of this chapter. This seems to be illogical but, in fact, one could go on repeating the diagrams of the chapter deriving better figures from each repetition and feeding these in to the next repetition. Such a process would serve no purpose here as, once again, it is the shape of the diagram which matters. Diagram 10-1 is a polar performance diagram. I have added an arrow to it to show the true wind speed. The diagram is drawn to scale using a true wind speed of 10 mph and a maximum boat speed of 3 mph. The true wind speed and the speed of the yacht are two vectors in a triangle of which the third is the apparent wind speed. Two sample triangles have been drawn. They could all be drawn except for the congestion of lines that would result.


At this early stage it is worth noting the obvious. The speed of the yacht is greatest when it is reaching. The speed drops considerably as it turns to go down wind and it also drops as the yacht turns upwind towards its course when beating close to the wind. This is typical for all yachts using the Bermuda rig. There must be good reasons for this and we have enough information to explain the diagram provided that we use the swing rig as our example.


The swing rig

The swing rig is a special version of the Bermuda rig that finds considerable use in model yachts. The main advantage of the swing rig is evident when it is running before the wind with the rig set as it is shown in Picture 10-2. All the sail area is always exposed directly to the wind. With a conventional rig there can be no guarantee of this as both sails can be on the same side and, on balance, the swing rig has the edge when running. The swing rig is not permitted on all the model classes but it is used extensively on Marbleheads. I have used a swing rig in chapter 9 to show flow patterns. The principal feature of the swing rig is that the whole rig swings with the mast in a socket in the hull as is evident from Pictures 10-2 and 10-3. The main boom is attached rigidly to the mast and the fore sail boom swivels on a second boom which is also rigidly attached to the mast. A single sheeting cord controls the rig through a sheeting post and a sheeting eye on the main boom. This arrangement eliminates the complexity of a second sheeting cord to control the fore sail. However, as a consequence, the sails have the same position relative to each other for all points of sailing.


It is this latter fact that permits us to consider the swing rig at this point in the text. The rig must be set up before it can be used and it is always set up for beating close to the wind. This means that the configuration of the sails that we looked at in chapter 9 is the only set up used in the swing rig. It follows that we are in a position to draw a diagram for a swing rig when beating and from this draw other diagrams for the other points of sailing. From these we can get an idea of how to think about the way that a yacht achieves its speed at all the points of sailing and how it can be set up and controlled.


Combined force on a swing rig

The fact that the rig swings on the hull and that the sails do not move relative to each other means that the rig should always work at the same angle to the apparent wind. It follows that the force exerted on the sails is sometimes almost across the course of yacht and at others nearly in line with its course. We do not know the magnitude of this force nor its direction relative to the rig but we do know that it will change with the magnitude of the apparent wind. As we can see from Diagram 10-1 even if the true wind is steady the direction and magnitude of the apparent wind changes with the speed and course of the yacht.


If we make some sensible simplifications we can draw diagrams from which useful information can be gathered. The first thing to do is to find a direction for the force on the rig. This means that we must put a direction to the force on each sail and, at least, a relative value to its magnitude. We have already decided that the force exerted on a sail is likely to be at right angles to the chord line and preferably inclined slightly towards the luff. Given this we can make a useful guess at the direction of the force on each sail. If now we suppose the forces to be proportional to the sail areas for any wind speed we can sort out a direction for the combined force acting on the rig.

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Diagram 10-4b Force diagram
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Diagram 10-4a Forces on a swing rig

I have drawn Diagrams 10-4a and 10-4b for a swing rig for a Marblehead. Typically the area of the fore sail is 40% of the area of the main sail. The fore sail boom is shown at 20°. The forces are drawn at right angles to the booms and they are drawn to lengths which are proportional to the areas. The two forces are then combined in the force diagram to give a resultant force on the rig. This force is exerted on the yacht through the mast and, via the sheeting cord, on the sheeting post. This force must act aft of the mast so that there is always a force in the sheeting cord. This fact has an influence on the choice of area for the fore sail. I have shown the position of the combined force fairly accurately. With the decisions above, the angle of the force is about 7° acting forwards relative to the boom.


Before we can draw sailing diagrams for a yacht with a swing rig we need to know something about the way in which this force varies with the change in the apparent wind. Let me explain.


For a sail that makes the same angle to the apparent wind for all points of sailing the force on the sail varies primarily with the square of the speed of the apparent wind. This speed is greatest when the yacht is beating and least when it is running before the wind. If we use the figure of 10 mph for the true wind speed and let the maximum boat speed be 3 mph the greatest speed of the apparent wind is 11.8 mph and the least is about 7.7 mph. The ratio of the squares of these speeds is 2.35 to 1. This means that any attempt to draw sailing diagrams must allow for changes in the force on the rig with changes in the speed of the apparent wind.


Before we draw diagrams it is necessary to recognise that there is a physical limitation to the sheeting mechanism of the swing rig. It comes when the rig reaches an angle of about 90° to the centre line of the yacht. For greater angles the force exerted by the rig on the sheeting may well take the rig over centre with permanent loss of control and possible damage to the sheeting gear. It follows that we can draw diagrams for sheeting out to 90° and then we must consider all other points of sailing with the rig in this same limiting position.


The sailing diagrams

Diagrams 10-5 are a set of 10 diagrams in which the true wind is always shown vertically. The course of the yacht is always the uppermost arrow. The force on the rig is shown in relative magnitude and direction pointing mainly downwards to the right and the component of this force in the direction of the course, i.e. the force driving the yacht, is shown between the two.


The velocity triangles give the magnitude and direction of the apparent wind for various courses starting from beating at 42.5° and then every 10° from 50° to 130°. In order to draw these I used the polar diagram given in Diagram 10-1 to find the boat speed.


For each course I have drawn the deck and the position of the sails. I have allowed for the leeway starting with a value of 3° and reducing it to zero as the transverse force decreases. In every diagram the angle of attack for the main sail is 32.5° to the apparent wind.


Inspection of the diagrams shows that, if one accepts the sheeting limit as being at 90° to the centre line, the course to which this corresponds is a little greater than 130°. This means that we must now decide what happens in the 50° between a course of 130° and running directly down wind.


The first thing to note is that, as no further movement of the rig relative to the yacht is possible, increasing the course angle towards running before the wind progressively increases the angle of attack of both sails. We have to decide whether the angle of the total force on the rig changes materially as a result of the increase in angle of attack. When we started with the swing rig we supposed that the force on each sail was at right angles to the chord of the sail. There seems to be no reason to think that this changes as the sail approaches an angle of attack of 90°. Furthermore we can continue to let the force on each sail be proportional to the sail area. If we
Diagrams 10-5 Sailing diagrams for a Swing Rig (for angles between 42.5° and 130°)















































































continue with this simplification then the direction of the total force is much the same as it is for the course angles up to 130°. As the course changes the speed of the apparent wind decays and the force decreases with the square of this speed. With these decisions the last four sailing diagrams can be drawn in the same way.


Diagrams 10-6 Sailing diagrams for yacht fitted with a Swing Rig (140° to 170°)

We now have a complete set of sailing diagrams and we must link these diagrams and the polar performance diagram.


Implications of the polar diagram

If we start with the diagram for beating close to the wind it is clear that the size of the component force available to drive the yacht is small compared with the total force on the rig yet the total force is its largest value. This small force has to overcome the air drag on the rig, on the deck fittings and on the hull above the water line as well as the water drag on the hull. We have seen in Chapter 7 Graphs 7-6 and 7-7 that a small force will drive a yacht without its rig at a speed that is a significant fraction of the maximum speed. It follows that provided that the rig and the deck are not cluttered with unnecessary and overlarge bits and pieces, the yacht will achieve a very useful speed. This is evident in the polar diagram where the speed when beating at 42.5° is about 65% of the maximum speed. This is really quite remarkable especially as most of the drive comes from the fore sail.


From beating through to sailing at 130° to the true wind the apparent wind falls steadily and in turn the combined force on the rig falls with the square of the apparent wind. However the rig turns relative to the yacht and progressively the combined force turns to become aligned with the centre line of the hull at 130°. The forward component of the combined force increases up to about 80° and then, by a quirk of the geometry, stays more or less the same right up to 130°. This can be seen in the resulting speeds of the yacht for this range of the polar diagram.


The last phase is that for the rig set at the limit of its sheeting when it can no longer turn to accommodate a change in course. Diagrams 10-6 show that the combined force is now very nearly along the centre line of the yacht[1]. The apparent wind falls appreciably as can be seen from the velocity diagrams. In turn the magnitude of the combined force falls off with the square of the apparent wind to give the low driving force (comparable to that for beating) and a speed when running down wind that is not much greater than that for beating.[2]


There remains one more piece of information for the sailor of a model racing yacht. If a yacht is to sail as fast as it can on a given point of sailing the rig must somehow be set to give the best drive. There is no problem with beating close to the wind; it is fully sheeted in (always assuming that it has been set up properly). Now we know that when running directly down wind or at angles up to 40° off the wind the sails should be fully sheeted out. In the 90° in between nothing is so obvious. It may be possible to use a masthead burgee to get an idea of the direction of the apparent wind and to set the rig accordingly but often the distance is too great for the burgee to be seen at all. The sails are the first guide, they must be filled and the fore sail not flapping and then the only other cue is the heeling of the yacht. This heeling is a response to the transverse force on the sails. Although the transverse forces have not been drawn on the sailing diagrams it is not difficult to see how big they are. Right up to about 110° we can attempt to set the rig by sheeting in until the heeling is a maximum with the sails filled. For the range of 110° to 140° the transverse force is very small and we can only look at the sails. Intuitive sailors do this without conscious effort. Ordinary people have to work at it.


[1] I suspect that this is oversimplified because the flow over the sails as they approach the run can become unstable and certainly very complicated. However the implication is correct.

[2] There is no escaping this poor performance down wind for a model yacht. Full sized yachts put up a spinnaker to increase the speed.